Recent studies including our own report (I) have revealed that heavily phosphorus (P) doped Czochralski-silicon (HP-Cz-Si) exhibits peculiar defect behaviors during crystal growth. HP-Cz-Si crystals with a low resistivity of around 0.6 mΩ cm (P concentration of 1.3 × 10^{20} P cm^{−3}) have interstitial-type stacking faults (SFs) and dislocations, which degrade device characteristics. The purpose of this paper is to clarify what causes the defect behavior in HP-Cz-Si through theoretical calculations. The thermal equilibrium concentrations of substitutional P (P_{s}), interstitial P (P_{i}), and (P_{s})* _{n}*-vacancy (

*V*) clusters (

*n*= 1−4) were determined by using density functional theory (DFT) calculations. The concentrations of P

_{i}([P

_{i}]) and (P

_{s})

*([(P*

_{n}V_{s)n}

*V*]) balanced with the given P

_{s}concentration ([P

_{s}]) were obtained as a function of the total P concentration ([P]) and the temperature. On the basis of the calculated results those can quantitatively explain our experimental results in the report (I), we propose a defect model that accurately represents HP-Cz-Si crystal growth. The main feature of the model is that the incorporated P

_{i}atoms at the solid/liquid interface around [P

_{i}] = 10

^{17}P

_{i}cm

^{−3}cause the formation of SFs and dislocations during the HP-Cz-Si crystal growth with around [P] = 10

^{20}P cm

^{−3}. Furthermore, DFT calculations were performed for P

_{i}segregation on the SF and for the photoelectron spectra of P 1s measured by hard x-ray photoelectron spectroscopy to explain the other experimental results in the report (I).

## I. INTRODUCTION

From the late 1970s to the early 2010s, many experimental studies have been conducted on the impact of dopants and impurities on intrinsic point defect behavior and grown-in defect formation during Si crystal growth.^{1–8} The types of dopants that have been reported are p-type (B and Ga), neutral (C, Ge, and Sn), and n-type (P, As, Sb, and Bi). The impacts of impurities (H, C, O, and N) have also been investigated for the purpose of improving the quality of Si crystals.^{4,9–15} Among the p-type and n-type dopants, B, P, and As can be doped at the highest concentration. The highest concentration of B, P, and As for which experimental results have been reported up until the early 2010s was about 1 × 10^{19} B cm^{−3}, 3.5 × 10^{19} P cm^{−3} and 3 × 10^{19} As cm^{−3}, respectively.^{4} Based on a model that quantitatively explains the intrinsic point defect behavior using density functional theory (DFT) calculations,^{16} the concentration distribution of intrinsic point defects valid for all pulling conditions in large-diameter Czochralski-Si (Cz-Si) crystal growth has been determined by computer simulation.^{17,18} Theoretical studies showed that the self-interstitial (*I*) and vacancy (*V*) formation energies around dopant atoms change depending on the type and size of the dopants, i.e., the electrical state and the local strain around the dopants. That is, the Si crystal becomes *I*-rich by B doping up to 1 × 10^{19} B cm^{−3} while it becomes *V*-rich by P doping up to 3.5 × 10^{19} P cm^{−3}and As doping up to 3 × 10^{19} As cm^{−3}.

In the last decade, heavily P doped (HP-Cz-Si) crystals have been widely used in power devices. To reduce the power consumption, the crystal resistivity must be reduced as much as possible. Currently, the most advanced crystals can be manufactured with a resistivity as low as 0.6 mΩ cm (1.3 × 10^{20} P cm^{−3}). The defect behavior in CZ-Si has been reported to change significantly when the P concentration in the Si crystal exceeds 3 × 10^{19} P cm^{−3}.^{4,19–26} That is, the void formation is significantly suppressed with P doping over (3–4) × 10^{19} P cm^{−3}.^{4,19} Senda *et al*.^{20} observed plate-like SiP^{21} precipitates of 100–200 nm in as-grown HP-CZ-Si crystals at around 8 × 10^{19} P cm^{−3}. Zeng *et al*.^{22,23} also reported the formation of oxygen precipitates from heterogeneous nuclei of small SiP precipitates during the heat treatment of HP-Cz-Si crystals. Subsequently, Wu *et al*.^{24} also reported that post-annealing of HP-CZ-Si crystals with a maximum [P] of 7.35 × 10^{19} P cm^{−3} at 450–1050 °C resulted in the formation of SiP precipitates of various crystallographic morphologies with dependent of temperature. Voronkov *et al.*^{25} and Nakamura *et al.*^{26} claimed that the increase of interstitial P (P_{i}) concentration causes the change of defect behavior from *V*-type to *I*-type in HP-Cz-Si.

Furthermore, in the 10^{20} P cm^{−3} order, we have identified even more peculiar defect behavior in the report (I).^{27} It was experimentally found that, at 1.3 × 10^{20} P cm^{−3}, small dislocation loops were observed in the bottom, while interstitial-type stacking faults (SFs) with P segregation were observed in the middle of crystal. The growing and tangling dislocations and P segregation were also observed in the crystal portion for a longer thermal history around 600 °C. The P segregation suggests the existence of supersaturated interstitial P_{i} atoms. However, no quantitative explanation of P_{i} concentration during HP-Cz-Si crystal growth has been given. Furthermore, various points still need to be clarified, such as the formation of SFs, the expansion of dislocations, and the actual state of defects in low-temperature regions during crystal growth of HP-Cz-Si.

In the present paper, the thermal equilibrium concentrations of substitutional P (P_{s}), interstitial P (P_{i}), and (P_{s})* _{n}V* (

*n*= 1−4) clusters in Si are obtained on the basis of DFT calculations. We propose an appropriate model of intrinsic point defect behavior in growing HP-Cz-Si. Furthermore, we perform DFT calculations for P

_{i}segregation on the SF and for the photoelectron spectra of P 1s measured by hard x-ray photoelectron spectroscopy (HAXPES) to explain the other experimental results in the report (I).

^{27}

## II. CALCULATION DETAILS

### A. Formation energy and formation entropy of P_{s}, P_{i}, and (P_{s})_{n}V clusters

_{n}V

DFT calculations were carried out within the generalized gradient approximation (GGA)^{28} for electron exchange and correlation using the CAmbridge Serial Total Energy Package (CASTEP) code.^{29} Three-dimensional periodic boundary conditions were set with cubic supercells of 512 Si atoms to calculate the total energy of Si crystals containing P_{s}, P_{i}, and (P_{s})* _{n}V* clusters. The cut-off energy of the plane waves was 340 eV. We carried out

*k*-point sampling at the Γ point. Note that if the model contains one P atom, its concentration [P] = 1 × 10

^{20}P cm

^{−3}is close to the actual concentration of the heavily P doped Si crystal. The formation energy of these P defects was obtained after geometry optimization.

_{i}atom which has the $ \u27e8 110 \u27e9$ dumbbell structure with a Si atom. The formation energies of the P

_{i}atom [

*E*(P

_{f}_{i})] and (P

_{s})

*clusters (*

_{n}V*E*((P

_{f}_{s})

*)) in Si were calculated with reference to the*

_{n}V*E*(P

_{f}_{s}) of uniformly distributed P

_{s}atoms in Si.

*E*of the isolated P

_{f}_{i}and P

_{i}up to 5th nearest substituted P

_{s}[Figs. 1(b)–1(f)] were obtained from

_{i}, one P

_{s}, and one P

_{i}at the

*ith*nearest P

_{s}, respectively.

*E*of the (P

_{f}_{s})

*clusters up to*

_{n}V*n*= 4 [Figs. 2(a)–2(d)] were obtained from

Here, $ E tot[ S i 511 \u2212 n ( P s ) n V]$ corresponds to the total energy of the cell including one $ ( P s ) nV$ cluster.

Due to the high calculation cost of the linear response method,^{30} formation (vibration) entropies of P_{s} atom [*S _{f}* (P

_{s})] and P

_{i}atom [

*S*(P

_{f}_{i})] were calculated using a cubic supercell of 64 Si atoms. The change of formation entropy through (P

_{s})

*cluster formation was not taken into consideration due to its low calculation accuracy of relatively large (P*

_{n}V_{s})

*cluster sizes for a 64-Si atom model.*

_{n}VThe thermal equilibrium concentrations of the P_{s}, P_{i}, and $ ( P s ) nV$ clusters were obtained on the basis of the calculated *E _{f}* and

*S*. The detail procedure will be described in Sec. III A.

_{f}### B. Energy reduction in P_{i} cluster growth on SF

An interstitial-type Frank-loop stacking fault (SF) was modeled as shown in Fig. 3 and compared with the perfect Si model. The SF model and perfect model consist of 32 and 30 (111) layers, respectively, including 16 atoms in each layer. To obtain the energy reduction of P_{i} at the SF with reference to the Si bulk, formation energy [*E _{f}* (P

_{i})] of the P

_{i}atom in Eq. (1a) was calculated by moving the position of the P

_{i}atom from the bulk to the SF.

By adding the other P_{i} atoms to the P_{i} atom on the SF, the most stable (P_{i})_{2}–(P_{i})_{7} clusters on the SF were obtained among the cluster structures considered. The energy reduction in P_{i} cluster growth on the SF was then calculated.

### C. Binding energy of P 1s electron for P_{s}, P_{i}, and (P_{s})_{n}V clusters

_{n}V

*E*) for P

_{b}_{s}, P

_{i}, and (P

_{s})

*clusters were calculated using the following formula:*

_{n}V^{31}

*N*− 1 electron system) and the ground state (the

*N*electron system), and $\Delta E core ( P )$ is defined as follows:

## III. RESULTS AND DISCUSSION

### A. Concentrations of P_{s}, P_{i}, and (P_{s})_{n}V clusters and mechanism of peculiar defect behaviors during HP-Cz-Si crystal growth

_{n}V

*E*), formation entropy (

_{f}*S*), site number (

_{f}*N*), and degeneracy number (

*w*) of the isolated P

_{i}_{s}atom, isolated P

_{i}atom, and P

_{i}atoms around the P

_{s}atom.

*E*and

_{f}*w*of $ ( P s ) nV$ clusters are included. Here, we considered only neutral defects. Experimental

_{i}^{32}and calculated

^{33}data from the literature are also included. The thermal equilibrium concentrations

*C*p

_{s}of P

_{s},

*C*p

_{i}of P

_{i}, and $ C ( P s ) n V of ( P s ) nV $ are calculated using the following formulas.

^{16,33}Note that

*C*p

_{i}depends on the given P

_{s}concentration [P

_{s}],

Here, *C*_{Si} = 5 × 10^{22} cm^{−3} is the site number of Si [= site number of P_{s} in Eq. (3a), site number of P_{i} in Eq. (3b), and site number of *V* in Eq. (3c)], *k* is the Boltzmann constant, and *T* is the temperature. The first row of Eq. (3b) indicates the *C*p_{i} of P_{i} atoms up to the 5th nearest substituted P_{s} atom, and the second row of Eq. (3b) indicates the *C*p_{i} of P_{i} atoms beyond the 5th nearest substituted P_{s} atom. Here, we assumed that *E _{f}* of the P

_{i}farther than the 5th nearest is not affected by the P

_{s}atom.

_{4}

*C*in Eq. (3c) is the degeneracy number of

_{n}*n*P

_{s}atoms at the nearest

*V*.

*E*= 0.4 eV for the P

_{f}_{s}atom in Eq. (4a) and the values of calculated changes of

*E*and

_{f}*S*summarized in Table I to obtain the ratios

_{f}*C*

_{Pi}/

*C*

_{Ps}and $ C ( P s ) n V/ ( C P s ) n.$

*C*

_{Pi}/

*C*

_{Ps}can be written by using Eqs. (3a) and (3b) as

*S*of P

_{f}_{i}does not change $[ S f i th ( P i ) = S f ( P i )]$ even with the interaction with P

_{s}. By the mass action law for the reaction of $ P s+I \u2190 \u2192 P i$, we obtained equilibrium P

_{i}concentration [P

_{i}] balanced to the given [P

_{s}] as the following formula,

Here, we assumed that the concentration of *I* maintained the thermal equilibrium concentration as the supersaturated *I* atoms were absorbed by the SFs and/or dislocation loops as will be mentioned later in this section.

^{33,34}we obtained the equilibrium $ ( P s ) nV$ concentration $[ ( P s ) n V]$ balanced to the given [P

_{s}] as the following formula:

Here, we also assumed that the concentration of *I* maintained the thermal equilibrium concentration.

^{tot}can be obtained by

*C*p

_{i}of P

_{i}was calculated using Eq. (3b). Figure 4 shows

*C*p

_{i}as a function of 1/

*T*for [P

_{s}] = 1 × 10

^{19}, 5 × 10

^{19}, 1 × 10

^{20}, and 2 × 10

^{20}P

_{s}cm

^{−3}. The dotted lines indicate the best fitted

*C*

_{pi}with an exponential function approximation. The

*C*

_{pi}can be summarized in one expression as

^{19}and 2 × 10

^{20}P

_{s}cm

^{−3}.

Figure 5 shows the thermal equilibrium concentrations, *C*p_{s} of P_{s} [Eq. (4a)], *C*p_{i} of P_{i} [Eq. (6)], and $ C ( P s ) n V of ( P s ) nV$ [Eqs. (7a)–(7d)] as a function of temperature. For *C*p_{i}, the values of [P_{s}] at 1 × 10^{19}, 5 × 10^{19}, 1 × 10^{20}, and 2 × 10^{20} P_{s} cm^{−3} are shown. Here, we assumed that the melting temperature *T* of HP-Cz-Si is 1412 °C. By using the thermal equilibrium concentration at the melting temperature, we obtained the incorporated P defect concentrations, [P_{s}] of P_{s}, [P_{i}] of P_{i}, and $[ ( P s ) n V]$ of $ ( P s ) nV$ as functions of total P concentrations [P]^{tot} as shown in Fig. 6(a). Here, we calculated [P_{i}] by Eq. (4c), $[ ( P s ) n V]$ by Eq. (4e), and [P]^{tot} by Eq. (5) at the given [P_{s}]. We clarified that the main P defects incorporated at the solid/liquid interface are P_{s} and P_{i} in the [P]^{tot} range of 1 × 10^{19}–2 × 10^{20} P cm^{−3}. Figure 6(b) shows the P defect concentrations, [P_{s}] of P_{s}, [P_{i}] of P_{i}, and $[ ( P s ) n V]$ of $ ( P s ) nV$ as functions of total P concentrations [P]^{tot} at *T* = 600 °C. Figure 6(c) shows P_{s}, P_{i}, and (P_{s})_{4}*V* concentrations at [P]^{tot} = 1 × 10^{20} P cm^{−3} as a function of temperature. The obtained expressions, *C*p_{s} [Eq. (4a)], *C*p_{i} [Eq. (6)], and $ C ( P s ) n V$ [Eqs. (7a)–(7d)], will be very impactful to expand the application of numerical simulation^{18} to HP-Cz-Si crystal growth. Here, we use these data in the following discussion on the mechanism of defect behaviors in HP-Cz-Si crystal growth.

Table II summarizes the incorporated P_{s} and P_{i} concentration at the melting temperature with their supersaturated temperatures. The data in Table II can be used quantitatively to explain our experimental results in the report (I)^{27} of the peculiar defect behavior around 10^{20} P cm^{−3} mentioned in Sec. I.

At the melting temperature, P_{i} around 10^{17} cm^{−3} is incorporated at the solid/liquid interface by heavily P doping around 10^{20} P cm^{−3}. *V* becomes supersaturated at a concentration of about 10^{14}–10^{15} *V* cm^{−3} after the pair recombination with *I*.^{4,35} P_{i} atoms become supersaturated at a concentration of about 10^{17} P_{i} cm^{−3} around void formation temperatures (around 1100 °C).^{36} Since the calculated diffusion barrier of P_{i} atom is very small (0.14 eV), the reaction P_{i} + *V *→ P_{s} proceeds, and the observable voids no longer form during crystal growth. Note that P_{i} still remains at [P_{i}] ∼ 10^{17} P_{i} cm^{−3} after the reaction P_{i} + *V *→ P_{s}.

During cooling of the crystal down to about 600 °C, the supersaturated P_{i} kicks out a lattice Si atom (P_{i} → P_{s} + *I*). The [P_{i}] is decreasing as can be seen from Fig. 6(c). The *I* becomes supersaturated and forms *I*-type dislocation loops and SFs.^{27} The concentration of *I* maintained the thermal equilibrium concentration as the supersaturated *I* atoms were absorbed by the SFs and/or dislocation loops.

At temperatures of 600 °C and lower, P_{s} becomes supersaturated. As can be seen from Figs. 6(b) and 6(c), the (P_{s})_{4}*V* concentration balanced to that of P_{s} is about 10^{19} P_{4}*V* cm^{−3} when [P]^{tot} = 10^{20} Pcm^{−3}. That is, a (P_{s})_{4}*V* cluster forms from the reaction 4P_{s} → (P_{s})_{4}*V* + *I* at 600 °C and lower. We will further discuss of the formation of (P_{s})_{4}*V* in Sec. III C. The generated *I* is absorbed by the *I*-type dislocation loops, causing defect growth and tangle. In addition, the P_{i} formed by P_{s} + *I *→ P_{i} segregates on SFs.^{27} The clustering of P_{i} atoms on SF will be discussed in Sec. III B. The formation of (P_{s})_{4}*V* drastically proceeds during long-time wafer annealing at 600 °C and lower.^{27}

Finally, we briefly discuss the case when [P]^{tot} is less than 5 × 10^{19} P cm^{−3}. As shown in Table II, P_{i} becomes supersaturated when below the void formation temperatures (around 1100 °C). Therefore, supersaturated *V* forms voids and later the reaction P_{i} → P_{s} + *I* occurs during crystal growth. This is probably the reason that voids were experimentally observed at [P]^{tot} less than 5 × 10^{19} P cm^{−3}.^{4}

### B. P_{i} segregation on SF

Figure 7 shows the most stable configuration of the P_{i} atom at the SF and the formation energy of the P_{i} atom with reference that on the SF. We found that the P_{i} atom becomes about 0.3 eV more stable when the P_{i} atom is trapped on the SF. The energy is not greatly reduced but it will be a sufficient driving force for trapping as the supersaturation of P_{i} increases as the temperature decreases during crystal growth.

_{i}atoms one by one to the P

_{i}atom on the SF, we discovered stable (P

_{i})

_{2}–(P

_{i})

_{7}clusters on the SF as shown in Fig. 8. Figure 9 shows the energy reduction

_{i}atom in the P

_{i}cluster growth on the SF. We found that the energy is reduced by about 0.8–2.0 eV per P

_{i}atom associated with the P

_{i}cluster growth. This result indicates that if one P

_{i}is trapped on the SF, the P

_{i}cluster growth will be drastic.

### C. Photoelectron spectra of P 1s measured by HAXPES

Table III summarizes the calculated binding energies of the P 1s electron (*E _{b}*) for P

_{s}, P

_{i}, and (P

_{s})

*clusters and energy shift Δ*

_{n}V*E*from the value of P

_{b}_{s}. The charge state determined by the energy band calculations is also shown. By comparing the calculated results with the experimental HAXPES results [P1, P2 (inactive) and P2 (active) peaks] in the report (I),

^{27}we concluded that the origin of the P1 peak should be P

_{s}. The origin of the P2 peak (inactive), which becomes noticeable after a longer thermal history at 600 °C and lower and/or long-time wafer annealing at 600 °C and lower, should be (P

_{s})

_{4}

*V*. (P

_{s})

_{2}

*V*is not the origin of the P2 peak (inactive) as its concentration is very low as shown in Fig. 6(b).

The P2 peak (active) was also observed in the as-grown crystal.^{27} Figure 10 shows two types of P_{3} clusters, (P_{s} + P_{i}) + P_{s}(a) and (P_{s} + P_{i}) + P_{s}(b), which are responsible for the P2 peak (active). These two clusters are stable as the energy of (P_{s} + P_{i}) + P_{s}(a) is reduced by −0.94 eV and that of (P_{s} + P_{i}) + P_{s}(b) is reduced by −1.10 eV from the isolated one P_{s} and two P_{i}. Furthermore, they have a +1 charge and Δ*E _{b}* close to that of the experimental results. That is, the two clusters are possible origins of P2 (active) formed during crystal growth.

## IV. CONCLUSION

HP-Cz-Si crystals with a resistivity down to 0.6 mΩ cm ([P] = 1.3 × 10^{20} P cm^{−3}) are currently being manufactured for application for low-voltage power MOSFETs. Recent studies including our own report (I)^{27} have revealed that HP-Cz-Si exhibits peculiar defect behaviors such as the formation of SFs and dislocations. The purpose of this paper is to clarify the causes of the defect behavior in HP-Cz-Si through theoretical calculations.

The thermal equilibrium concentrations of P_{s}, P_{i}, and (P_{s})* _{n}V* (

*n*= 1−4) clusters were determined by DFT calculations. Furthermore, equilibrium concentrations of P

_{i}and (P

_{s})

*balanced to the given P*

_{n}V_{s}concentration were obtained as functions of the total P concentration and the temperature. On the basis of the calculated results those can quantitatively explain our experimental results in the report (I),

^{27}we proposed the following defect model to represent HP-Cz-Si crystal growth. At the melting temperature, P

_{i}around 10

^{17}cm

^{−3}is incorporated at the solid/liquid interface by heavily P doping around 10

^{20}P cm

^{−3}. From 1100 to 600 °C, supersaturated P

_{i}atom interacts with the Si atom to become P

_{s}with the emission of

*I*. The emitted

*I*

_{s}agglomerate and form SFs and dislocations. At temperatures of 600 °C and lower, supersaturated P

_{s}becomes (P

_{s})

_{4}

*V*with the emission of

*I*. The SFs and dislocations absorb

*I*

_{s}and become more complex defects. The supersaturated P

_{i}also segregates on the defects. The formation of (P

_{s})

_{4}

*V*further proceeds during long-time wafer annealing at 600 °C and lower.

The other DFT calculations explained the P_{i} segregation on the SF. The calculated energy is reduced by about 0.8–2.0 eV for each P_{i} associated with the growth of a P_{i} cluster up to (P_{i})_{7}. That is, if one P_{i} is trapped on the SF, the growth of the P_{i} cluster will proceed drastically.

Finally, the binding energies of the P 1s electron for P_{s}, P_{i}, and (P_{s})* _{n}V* clusters were calculated and compared to the experimental HAXPES results [P1, P2 (inactive) and P2 (active) peaks] in the report (I).

^{27}We concluded that the origin of the P1 peak should be P

_{s}, and the origin of the P2 peak (inactive), which becomes noticeable after longer thermal history at 600 °C and lower and/or long-time wafer annealing at 600 °C and lower, should be (P

_{s})

_{4}

*V*. Lastly, we proposed two structures of the P

_{3}cluster responsible for the P2 peak (active) observed in the as-grown crystal.

## ACKNOWLEDGMENTS

This work was partially supported by JST, CREST, Japan (Grant No. JPMJCR21C2).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Koji Sueoka:** Conceptualization (lead); Data curation (lead); Formal analysis (lead); Funding acquisition (lead); Investigation (lead); Methodology (lead); Supervision (lead); Validation (lead); Writing – original draft (lead); Writing – review & editing (lead). **Yasuhito Narushima:** Investigation (supporting); Methodology (supporting); Validation (supporting). **Kazuhisa Torigoe:** Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal). **Naoya Nonaka:** Investigation (supporting); Methodology (supporting); Validation (supporting). **Koutaro Koga:** Investigation (supporting); Methodology (supporting); Validation (supporting). **Toshiaki Ono:** Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal). **Hiroshi Horie:** Formal analysis (equal); Investigation (supporting); Methodology (supporting); Validation (supporting). **Masataka Hourai:** Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Validation (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available within the article.

## REFERENCES

*The 78th JSAP Autumn Meeting*(The Japan Society of Applied Physics, 2017), 7p-PB6-5 (in Japanese).

*The 80th JSAP autumn meeting*(The Japan Society of Applied Physics, 2019), 18a-C212-5 (in Japanese).

*Crystal Growth and Characterization of Advanced Materials*